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ACTSC 445
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Jiahua Chen
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Actuarial Science

ACTSC 445

Jiahua Chen

Fall

Description

ACTSC 445: Asset-Liability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 6 Immunization
References (recommended readings): Chap. 3 of Financial Economics (on reserve at the library: call
number HG174 .F496 1998).
What is immunization?
Redington (1952): Immunization implies the investment of assets in such a way that existing
business is immune to a general change in the rate of interest.
Fisher-Weil (1971): A portfolio of investment is immunized for a holding period if its value at
the end of the holding period, regardless of the course of rates during the holding period, must be
at least as it would have been had the interest rate function been constant throughout the holding
period.
Implication: If the realized return on an investment in bonds is sure to be at least as large as the
appropriately computed yield to the horizon, then that investment is immunized.
An immunization strategy is a risk management technique designed to ensure that for any
small change in a specied parameter, a portfolio of debt instruments (e.g., T-bills, bonds, GICs
etc) will cover a liability (or liabilities) coming due at a future date (or over a period in the future).
It is a passive management technique because it takes prices as given and then tries to control
the risk appropriately. (By contrast, active management techniques try to exploit changes in (1)
the level of interest rates, (2) the shape of the yield curve (3) yield spreads, by using interest rate
forecasts and identication of mispriced bonds)
asset allocation problem (i.e., must choose assets that will produce an immunized portfolio)
Single-liability case
Well start with the case where there is only one liability in the portfolio, with corresponding cash ow
of Ltat some time t.
The goal is to choose an asset cash ow sequence {A ,t t 0} that will, along with L , protuce an
immunized portfolio. Lets start with an example.
Example I: Suppose an insurance company faces a liability obligation of $1 million in 5 years. The
available market instruments are: 3-year, 5-year and 7-year zero-coupon bonds, each yielding 6% annual
eective rate.
Portfolio A: Invest $747,258.17 in the 5-year zero coupon bond
1 Portfolio B: Invest the same amount (i.e. $747,258.17) in a 3-year zero coupon bond. The
maturity value at t = 3 is $889,996.44.
Portfolio C: Invest $747,258.17 in a 7-year zero coupon bond. The maturity value at t = 7 is
$1,123,600.00.
If the yields remain unchanged, then the 3 portfolios have the same value of $1 000 000 at time 5.
To verify if these portfolios are immunized or not, we need to look at what happens if, immediately after
the portfolio is acquired, the yield changes instantaneously and remains constant at that level.
First, note that for portfolio A, this change has no impact: its value at time 5 is still $1 000 000. But
this is not true for portfolios B and C, as Tables 1 and 2 show.
Table 1: Value of Portfolio B for dierent yields
Value of Portfolio B Capital Gain Implied
(%) at time 0 at time 5 at time 0 Yield (%)
4.00 791203.5944 962620.1495 43945.4215 5.20
5.00 768812.3874 981221.0751 21554.2146 5.60
5.90 749377.0511 998114.0975 2118.8782 5.96
6.00 747258.1729 1000000.0000 0.0000 6.00
6.10 745147.2753 1001887.6824 2110.8975 6.04
7.00 726502.2044 1018956.9242 20755.9684 6.40
8.00 706507.8685 1038091.8476 40750.3044 6.80
So for portfolio B, if the yields go up, then we realize a gain at time 5, because we can reinvest the
proceeds obtained at time 3 at a high yield. But if the rates drop, then we realize a loss at time 5. The
problem here is the reinvestment risk.
Table 2: Value of Portfolio C for dierent yields
Value of Portfolio C Capital Gain Implied
(%) in year 0 at time 5 at time 0 Yield (%)
4.00 853843.6549 1038831.3609 106585.4820 6.81
5.00 798521.5425 1019138.3220 51263.3697 6.40
5.90 752211.5711 1001889.4658 4953.3982 6.04
6.00 747258.1729 1000000.0000 0.0000 6.00
6.10 742342.0181 998115.8742 4916.1548 5.96
7.00 699721.6100 981395.7551 47536.5629 5.60
8.00 655609.8081 963305.8985 91648.3647 5.21
The situation here is opposite from what we face with Portfolio B: if the rates drop, then we can sell
the 7-year zero bond at a higher price at time 5, which results in a gain. But a yield increase produces
a loss. The problem here is the interest rate or price risk.
2 Observations from Example I
With a single liability, the best immunization strategy is the one for which the asset cash ow
coincides with the liability cash ow
When asset cash ows occur prior to (or after) the liability cash ow, the portfolio is subject
to reinvestment risk (or market/interest rate/price risk).
A valid question is: could we construct a portfolio containing cash ows occuring before and after the
liability due date that could be immunized? Motivation:
Any initial capital loss may be oset in time by greater returns from reinvestment.
Similarly, any initial capital gain may be oset in time by lower returns from reinvestment.
Does there exist an optimum trade-o? I.e., a way to construct a portfolio like this that
maximizes (in some sense) the gain?
The following example studies this idea.
Example II: Consider Portfolio D, which consists in an investment of $373629.0864 in 3-year zero-
coupon bonds, and $373629.0864 in 7-year zero-coupon bonds. Their maturity values are, respectively,
444,998.22 and 561,800.00. Note that the Macaulay duration of this portfolio is 5.
If the yields remain unchanged, then at t = 5 we have 373,629.0864 2 (1.06) = 1000000.
If the rates change, then we get the following results:
Value of Portfolio D Capital Gain Implied
(%) at time 0 at time 5 at time 0 Yield (%)
4.0 822523.6247 1000725.7552 75265.4518 6.01538
5.0 783666.9650 1000179.6986 36408.7921 6.00381
5.9 750794.3111 1000001.7817 3536.1382 6.00004
6.0 747258.1729 1000000.0000 0.0000 6.00000
6.1 743744.6467 1000001.7783 -3513.5262 6.00004
7.0 713111.9072 1000176.3396 -34146.2657 6.00374
8.0 681058.8383 1000698.8731 -66199.3346 6.01481
Hence with this portfolio, a gain is realized at time 5 for all alternative yi considered...
Note that at time 0, there is a capital loss for portfolio D. More generally, we can look at the value of
this portfolio at time t if the initial yield goes from 6%. That is, we can consider the value
V = 444998.22(1 + y )(3t)+ 561800.00(1 + y)(7t)
t
for t = 1,...,10 and dierent s.
3 if rate drops y if rate rises
t 4.00% 5.50% 5.90% 6.00% 6.10% 6.50% 8.00%
0 822524 765169 750794 747258 743745 729913 681059
1 855425 807253 795091 792094 789113 777358 735544
2 889642 851652 842002 839619 837249 827886 794387
3 925227 898493 891680 889996 888321 881698 857938
4 962236 947910 944289 943396 942509 939009 926573
5 1000726 1000045 1000002 1000000 1000002 1000044 1000699
6 1040755 1055047 1059002 1060000 1061002 1065047 1080755
7 1082385 1113075 1121483 1123600 1125723 1134275 1167215
8 1125680 1174294 1187650 1191016 1194392 1208003 1260592
9 1170708 1238880 1257722 1262477 1267250 1286523 1361440
10 1217536 1307018 1331927 1338226 1344552 1370147 1470355
Equivalently, we can look at the corresponding implied yield for the portfolio, which is the value i such
that
747,258.17 = V (1 + i)t.
t
if rate drops y if rate rises
t 4.00% 5.50% 5.90% 6.00% 6.10% 6.50% 8.00%
1 14.48 8.03 6.40 6.00 5.60 4.03 1.57
2 9.11 6.76 6.15 6.00 5.85 5.26 3.11
3 7.38 6.34 6.07 6.00 5.93 5.67 4.71
4 6.53 6.13 6.03 6.00 5.98 5.88 5.52
5 6.02 6.00 6.00 6.00 6.00 6.00 6.01
6 5.68 5.92 5.98 6.00 6.02 6.08 6.34
7 5.44 5.86 5.97 6.00 6.03 6.14 6.58
8 5.26 5.81 5.96 6.00 6.04

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